Math 220, Sections 201/202—Homework #5
due Wednesday, February 23, 2005
Remember that all of your solutions must be written in complete sentences that are easy
to read and in logically correct order.
I. D’Angelo and West, p. 95, #4.11
II. Find, with proof, a bijection f from (0, 1] to [2, 4). What is the inverse of f ?
III. Let a, b, and c be real numbers, and define a function f : R → R by f ( x) =
ax2 + bx + c.
(a) If a 6= 0, prove that f is not bijective. (Possible hint: complete the square.)
(b) If a = 0 and b 6= 0, prove that f is bijective.
IV. For each of the parts below, either write down a specific function with the specified
properties (no proof necessary), or explain why such a function does not exist.
(a) a : N → N such that a is injective but not surjective
(b) b : N → N such that b is surjective but not injective
(c) c : N → N such that c is a bijection
(d) d : {1, 2, 3, 4} → {1, 2, 3, 4, 5, 6} such that d is a surjection
(e) e : {1, 2, 3, 4} → {1, 2, 3, 4, 5, 6} such that e is not an injection
(f) f : {1, 2, 3, 4, 5} → {1, 2, 3, 4, 5} such that f is surjective but not injective
V. D’Angelo and West, p. 95, #4.12
VI. (a) Let f : [0, 1] → R be a decreasing function. Prove that f is bounded.
(b) Let g : (0, 1) → R be a decreasing function. Give an example to show that g is
not necessarily bounded.
(c) Let h : R → R be a bounded function. Prove that h is not surjective.